NODAL Vector solutions with clustered peaks for a nonlinear elliptic equations in $\R^3$

In this paper, we study the following coupled nonlinear Schrödinger system in $\R^3$ $$ \left\{% \begin{array}{ll} -ε^2Δu +P(x)u=μ_1 u^3+βv^2u,~~&x\in \R^3,\vspace{0.15cm}\\ -ε^2Δv +Q(x)v=μ_2 v^3+βu^2v,~~&x\in \R^3,\\ \end{array}% \right. $$ where $μ_1 >0,μ_2>0$ and $β\in \R$ is a coupling constant. Whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small $ε$ under some additional conditions on $P(x), Q(x)$ and $β$. Moreover, the number of this type of solutions will go to infinity as $ε\to 0^+$.

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